Analytical theory for the initial mass function: III time dependence and star formation rate

TL;DR

This paper extends the analytical theory of the stellar initial mass function (IMF) by incorporating time dependence and magnetic fields, providing new insights into star formation rates and the evolution of the IMF.

Contribution

It introduces a time-dependent analytical model for the IMF and star formation rate, accounting for magnetic fields and clump evolution, aligning well with observations and simulations.

Findings

Mass spectra are similar to previous models but with shallower slopes at high masses.

The star formation rate depends on clump mass/size, explaining observed scatter.

The model reproduces features seen in numerical simulations of converging flows.

Abstract

The present paper extends our previous theory of the stellar initial mass function (IMF) by including the time-dependence, and by including the impact of magnetic field. The predicted mass spectra are similar to the time independent ones with slightly shallower slopes at large masses and peak locations shifted toward smaller masses by a factor of a few. Assuming that star-forming clumps follow Larson type relations, we obtain core mass functions in good agreement with the observationally derived IMF, in particular when taking into account the thermodynamics of the gas. The time-dependent theory directly yields an analytical expression for the star formation rate (SFR) at cloud scales. The SFR values agree well with the observational determinations of various Galactic molecular clouds. Furthermore, we show that the SFR does not simply depend linearly on density, as sometimes claimed in the literature, but depends also strongly on the clump mass/size, which yields the observed scatter. We stress, however, that {\it any} SFR theory depends, explicitly or implicitly, on very uncertain assumptions like clump boundaries or the mass of the most massive stars that can form in a given clump, making the final determinations uncertain by a factor of a few. Finally, we derive a fully time-dependent model for the IMF by considering a clump, or a distribution of clumps accreting at a constant rate and thus whose physical properties evolve with time. In spite of its simplicity, this model reproduces reasonably well various features observed in numerical simulations of converging flows. Based on this general theory, we present a paradigm for star formation and the IMF.